BGE Animated Fireball Group
Groups within the BGE

Typically you do not work with groups as the BGE does not support groups.

The BGE supports group instances, but a bit differently than Blender.

An instance typically consists of 


the instance members (copies of the objects assigned to the group) and 
the dupli group objects (typically an empty)


While in Blender you can move the dupli group object around and the members will follow within the BGE they have no relationship. When you move the dupli group object the instance member will remain where they are.

I just tell you that to explain why I think your issue has nothing to do with groups.

Orbiting animation

Orbiting usually occurs when you move an object forward along a local axis and rotate at a local axis. You basically change the direction of your forward move. In natural words: you move in a curve

Spinning is orbiting without moving.

But how to move along a path and still spin around yourself? You do not move forward at you local axis, but at another axis that is not local. Example: you do a step towards your target without caring your local orientation. 

This can be hard as you need to compensate your local orientation.

What if you do not move at all? You let you move and just focus on orbiting.

A simple way is you parent the spinning object to another object (e.g. an empty). This way the spinning object does not move (relative to the parent). Then you can move the parent as you like. The motion will not influence the rotation and vice versa. Make sure you move the parent and turn the child.